A magnetic dipole of moment sits in a uniform field at angle to the field

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Variational and asymptotic methods

Mathematical Methods for Physics, Ch Series, limits and controlled approximation

Problem

A magnetic dipole of moment mm sits in a uniform field BB at angle θ\theta to the field. Its potential energy is U(θ)=−mBcos⁡θU(\theta) = -mB\cos\theta. For small θ\theta this is often replaced by the two-term approximation U≈−mB(1−θ22)U \approx -mB\left(1 - \dfrac{\theta^2}{2}\right). (a) Using the alternating series bound on the cosine series, bound the size of the error in U/(mB)U/(mB) (that is, the size of cos⁡θ−(1−θ2/2)\cos\theta - (1-\theta^2/2)) at θ=0.40 rad\theta = 0.40\,\mathrm{rad}. (b) Compute the exact error cos⁡θ−(1−θ2/2)\cos\theta - \left(1-\theta^2/2\right) at θ=0.40 rad\theta = 0.40\,\mathrm{rad} and check it lies below the bound found in part (a).

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